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Page No 10.14:

Question 1:

In the given figure, OA and OB are opposite rays:

(i) If x = 25°, what is the value of y?

(ii) If y = 35°, what is the value of x?

Answer:

In figure:

Since OA and OB are opposite rays. Therefore, AB is a line. Since, OC stands on line AB.

Thus,and form a linear pair, therefore, their sum must be equal to.

Or, we can say that

From the given figure:

and

On substituting these two values, we get

...(i)

(i) On puttingin (i), we get:

Hence, the value of y is.

(ii) On putting in in equation (A), we get:

Hence, the value of x is.

Page No 10.14:

Question 2:

In the given figure, write all pairs of adjacent angles and all the linear pairs.

Answer:

The figure is given as follows:

The following are the pair of adjacent angles:

and

and

The following are the linear pair:

and

and

Page No 10.14:

Question 3:

In the given figure, find x. Further find ∠BOC, ∠COD and∠AOD

Answer:

In the given figure:

AB is a straight line. Thus,, and form a linear pair.

Therefore their sum must be equal to.

We can say that

(i)

It is given that, and.

On substituting these values in (i), we get:

It is given that:

Therefore,

Also,

Therefore,

Therefore,

Page No 10.14:

Question 4:

In the given figure, rays OA, OB, OC, OD and OE have the common end point O. Show that ∠AOB + ∠BOC + ∠COD + ∠DOE + ∠EOA = 360°.

Answer:

Let us draw a straight line.

,and form a linear pair. Thus, their sum should be equal to.

Or, we can say that:

(I)

Similarly,,and form a linear pair. Thus, their sum should be equal to.

Or, we can say that:

(II)

On adding (I) and (II), we get:

Hence proved.

Page No 10.15:

Question 5:

In the given figure, ∠AOC and ∠BOC form a linear pair. If a − 2b = 30°, find a and b.

Answer:

In the figure given below, it is given thatand forms a linear pair.

Thus, the sum of and should be equal to.

Or, we can say that:

From the figure above, and

Therefore,

It is given that:

On comparing (i) and (ii), we get:

Putting in (i), we get :

Hence, the values for a and b areand respectively.

Page No 10.15:

Question 6:

How many pairs of adjacent angle are formed when two lines intersect in a point?

Answer:

Suppose we have two lines, say AB and CD intersect at a point, O as shown in the figure below:

Then there are 4 pairs of adjacent angles formed, namely:

and

and

and

and

Page No 10.15:

Question 7:

How many pairs of adjacent angles, in all, can you name in the given figure.

Answer:

In the given figure,

We have 10 adjacent angle pairs, namely:

and

and

and

and

and

and

and

and

and

and

Page No 10.15:

Question 8:

In the given figure, determine the value of x.

Answer:

In the given figure:

is a straight line. Thus,and form a linear pair.

Therefore their sum must be equal to.

We can say that

It is given that, substituting this value in equation above, we get:

Page No 10.15:

Question 9:

In the given figure, AOC is a line, find x.

Answer:

It is given that AOC is a line. Therefore, and form a linear pair. Thus, the sum of and must be equal to .

Page No 10.16:

Question 18:

(i) Find the measure of ∠FOE, ∠COB and ∠DOE.
(ii) Name all the right angles.
(iii) Name three pairs of adjacent complementary angles.
(iv) Name three pairs of adjacent supplementary angles.
(v) Name three pairs of adjacent angles.

Answer:

The given figure is as follows:

(i)

It is given that,,and form a linear pair .

Therefore, their sum must be equal to .

That is ,

It is given that :

,

and

in equation above, we get:

It is given that:

From the above figure:

Similarly, we have:

From the above figure:

(ii)

We have:

From the figure above and the measurements of the calculated angles we get two right angles as and.

Two right angles are already given asand.

(iii)

We have to find the three pair of adjacent complementary angles.

We know that is a right angle.

Therefore,

and are complementary angles.

Similarly, is a right angle.

Therefore,

and are complementary angles.

Similarly, is a right angle.

Therefore,

and are complementary angles.

(iv)

We have to find the three pair of adjacent supplementary angles.

Since,is a straight line.

Therefore, following are the three linear pair, which are supplementary:

and ;

and and

and

(v)

We have to find three pair of adjacent angles, which are as follows:

and

and

and

Page No 10.17:

Question 19:

In the given figure, POQ is a line. Ray OR is perpendicular to line PQ.OS is another ray lying between rays OP and OR. Prove that ∠ROS = 12 (∠QOS − POS).

Answer:

The given figure is as follows:

We have POQ as a line. Ray OR is perpendicular to line PQ. Therefore,

From the figure above, we get:

(i)

and form a linear pair. Therefore,

(ii)

From (i) and (ii) equation we get:∠QOS+∠POS=2×90

Hence proved.

Page No 10.22:

Question 1:

In the given figure, lines l1 and l2 intersect at O, forming angles as shown in the figure. If x = 45, find the values of y, z and u.

Answer:

It is given that lines and intersect at a point.

Therefore,and are the two linear pairs are formed.

Thus,

Also,

It is given that, putting this value above, we get:

Also we have a two pairs of vertically opposite angles in the figure, that is,and .

We know that, if two lines intersect, then the vertically opposite angles are equal.

Thus,

And

Page No 10.22:

Question 2:

In the given figure, three coplanar lines intersect at a point O, forming angles as shown in the figure. Find the values of x, y, z and u.

Answer:

It is given that the lines, and intersect at a point.

Therefore, vertically opposite angles should be equal

Also, ,and form a linear pair.

Therefore,

Substituting ,and in equation above:

andare vertically opposite angles.

Therefore,

Substituting , in equation above:

Similarly, ,are vertically opposite angles.

Therefore,

Substituting , in equation above:

Page No 10.22:

Question 3:

In the given figure, find the values of x, y and z.

Answer:

In the given question, the values of x, y, and z will be determined as follows:z and 25° form a linear pair.​So,z+25°=180°⇒z=180-25⇒z=155°

Now, z and x are vertically opposite to each other. So, x = 155°.

Also, y and x form a linear pair. ​So,y+155°=180°⇒y=180-155⇒y=25°

Hence, the values are x=155°,y=25°andz=155°.

Page No 10.22:

Question 4:

In the given figure, find the value of x.

Answer:

In the following figure we have to find the value of x

In the figure AB, CD and EF are lines; therefore, angles COF and EOD are vertically opposite angles.

Therefore,

Since, AB is a straight line, so

Hence, .

Page No 10.22:

Question 5:

If one of the four angles formed by two intersecting lines is a right angle, then show that each of the four angles is a right angle.

Answer:

The given problem can be drawn as :

It is given that

Also,and form a linear pair.

Therefore, their sum must be equal to.

Substituting, above, we get:

Similarly, we can prove that

and

Hence, we have proved that ,If one of the four angles formed by two intersecting lines is a right angle, then show that each of the four angles is a right angle.

Page No 10.23:

Question 6:

In the given figure, rays AB and CD intersect at O.

(i) Determine y when x = 60°

(ii) Determine x when y = 40

Answer:

Raysand intersect at point.

Therefore, and form a linear pair.

Thus,

(i)

On substituting:

(ii)

On substituting:

Page No 10.23:

Question 7:

In the given figure, lines AB, CD and EF intersect at O, Find the measure of ∠AOC, ∠COF, ∠DOE and ∠BOF.

Answer:

It is given thatand intersect at a point

Thus and are vertically opposite angles, therefore, these must be equal.

That is,

Similarly,and intersect at a point.

Thusand are vertically opposite angles, therefore, these must be equal.

That is,

Similarly,and intersect at a point.

Thusand are vertically opposite angles, therefore, these must be equal.

That is,

Also,,and form a linear pair. Therefore, their sum must be equal to .

Putting in (I):

Page No 10.23:

Question 8:

AB, CD and EF are three concurrent lines passing through the point O such that OF bisects ∠BOD. If ∠BOF = 35°, find ∠BOC and ∠AOD.

Answer:

The corresponding figure is as follows:

Three concurrent lines are given as follows:

AB,CD and EF

Also, OF is the bisector of and it is given that.Therefore,

Also,

Since, andare vertically opposite angles. Therefore,

From (i) equation:

We know that and form a linear pair.

Thus,

Similarly, and form a linear pair.

Thus,

Page No 10.23:

Question 9:

In the given figure, lines AB and CD intersect at O. If ∠AOC + ∠BOE = 70° and ∠BOD = 40°, find ∠BOE and reflex ∠COE.

Answer:

In the figure, ,and form a linear pair.

Thus,

It is given that, on substituting this value, we get:

Thus, reflex

Therefore, reflex

Sinceand are vertically opposite angles, thus, these two must be equal.

Therefore,

But, it is given that :

Substituting in above equation:

Page No 10.23:

Question 10:

Which of the following statements are true (T) and which are false (F)?

(i) Angles forming a linear pair are supplementary.
(ii) If two adjacent angles are equal, then each angle measures 90°.
(iii) Angles forming a linear pair can both be acute angles.
(iv) If angles forming a linear pair are equal, then each of these angles is of measure 90°.

Answer:

(i) True

As the sum of the angles forming a linear pair is.

(ii) False

As the statement is incomplete in itself.

(iii) False

Let us assume one of the angle in a linear pair be; such that ,that is, an acute angle.

Therefore, the other angle in the linear pair becomes, which clearly cannot be acute.

(iv) True

Let one of the angle in the linear pair be. Then, other angle also becomes equal to.

Therefore, by the definition of linear pair, we get:

.

Hence, if angles forming a linear pair are equal, then each of these angles is of measure.

Page No 10.23:

Question 11:

Fill in the blanks so as to make the following statements true:

(i) If one angle of a linear pair is acute, then its other angle will be ........
(ii) A ray stands on a line, then the sum of the two adjacent angles so formed is ..........
(iii) If the sum of two adjacent angles is 180°, then the ........ arms of the two angles are opposite rays.

Answer:

(i)

If one angle of a linear pair be acute, then its other angle will be obtuse.

Explanation:

Let us assume one of the angle in a linear pair be; such that,that is, an acute angle.

Therefore, the other angle in the linear pair becomes, which clearly cannot be acute.

(ii)

A ray stands on a line, and then the sum of the two adjacent angles so formed is.

Explanation:

The statement talks about two adjacent angles forming a linear pair.

(iii) If the sum of the two adjacent angles is, then the uncommon arms of the two angles are opposite rays.

Explanation:

The statement talks about two adjacent angles forming a linear pair.

Therefore, this can be drawn diagrammatically as:

Page No 10.23:

Question 12:

Prove that the bisectors of a pair of vertically opposite angles are in the same straight line.

Answer:

Let AB and CD intersect at a point O

Also, let us draw the bisectors OP and OQ of and.

Therefore,

And

We know that,and are vertically opposite angles. Therefore, these must be equal, that is:

We know that:

From (i)

From (ii)

This means, , and form a linear pair.

Hence, POQ forms a straight line.

Thus, we can say that the bisectors of a pair of vertically opposite angles are in the same straight line.

Page No 10.23:

Question 13:

If two straight lines intersect each other, prove that the ray opposite to the bisector of one of the angles thus formed bisects the vertically opposite angle.

Answer:

Let AB and CD intersect at a point O

Also, let us draw the bisector OP of .

Therefore,

Also, let’s extend OP to Q.

We need to show that, OQ bisects.

Let us assume that OQ bisects, now we shall prove that POQ is a line.

We know that,

and are vertically opposite angles. Therefore, these must be equal, that is:

and are vertically opposite angles. Therefore,

Similarly,

We know that:

Thus, POQ is a straight line.

Hence our assumption is correct. That is,

We can say that if the two straight lines intersect each other, then the ray opposite to the bisector of one of the angles thus formed bisects the vertically opposite angles.

Page No 10.46:

Question 1:

In the given figure, ABCD and ∠1 and ∠2 are in the ratio 3:2 Determine all angles form 1 to 8.

Answer:

The given figure is as follows:

It is give that the lines AB and CD are parallel and angles 1 and 2 are in the ratio 3: 2.

Let

In the figure angle 1 and 2 are supplementary. So,

3x + 2x= 180

⇒ 5x= 180

⇒ x = 36∠1=36×3=108°and∠2=36×2=72°

Since, angle 1 and 5 and angle 2 and 6 are corresponding angles, so

Since, angles 1 and 3 and 2 and 4 are vertically opposite angles, so

Now,

Angle 5 and 6 and angle 6 and 8 are vertically opposite angles, so

Hence,and.

Page No 10.46:

Question 2:

In the given figure, l, m and n are parallel lines intersected by transversal p at X, Y and Z respectively. Find ∠1, ∠2 and ∠3.

Answer:

According to the given figure,m || n and are cut by transversal p. ∠2=120°(alternateinterioranglesareequal)
Also, l || m. So, ∠1=∠3(correspondingangles)
Also, ∠3and120°formalinearpair.∠3+120°=180°⇒∠3=180-120⇒∠3=60°

And ∠1=∠3=60°,∠2=120°

Page No 10.46:

Question 3:

In the given figure, if AB || CD and CD || EF, find ∠ACE.

Answer:

The figure is given as follows:

It is given that AB || CD and CD || EF

Thus,and are alternate interior opposite angles.

Therefore,

Also, we have

From the figure:

From equations (i) and (ii):

Hence, the required value for is.

Page No 10.46:

Question 4:

In the given figure, state which lines are parallel and why.

Answer:

The given figure is as follows:

Since

These are the pair of alternate interior opposite angles.

Theorem states: If a transversal intersects two lines in such a way that a pair of alternate interior angles is equal, then the two lines are parallel.

Therefore,

Page No 10.47:

Question 5:

In the given figure, if l || m, n || p and ∠1 = 85°, find ∠2.

Answer:

The figure is given as follows:

It is given that .

Thus,and are corresponding angles.

Therefore,

It is given that . Therefore,

...(i)

Also, we have .

Thus,and are consecutive interior angles.

Therefore,

From equation (i), we get:

Hence, the required value for is .

Page No 10.47:

Question 6:

If two straight lines are perpendicular to the same line, prove that they are parallel to each other.

Answer:

The figure can be drawn as follows:

Here, and.

We need to prove that

It is given that , therefore,

(i)

Similarly, we have , therefore,

(ii)

From (i) and (ii), we get:

But these are the pair of corresponding angles.

Theorem states: If a transversal intersects two lines in such a way that a pair of corresponding angles is equal, then the two lines are parallel.

Thus, .

Page No 10.47:

Question 7:

Two unequal angles of a parallelogram are in the ratio 2 : 3. Find all its angles in degrees.

Answer:

The parallelogram can be drawn as follows:

It is given that

Therefore, let:

and

We know that opposite angles of a parallelogram are equal.

Therefore,

Similarly

Also, if , then sum of consecutive interior angles is equal to .

Therefore,

We have

Also,

Similarly,

And

Hence, the four angles of the parallelogram are as follows:

, , and .

Page No 10.47:

Question 8:

In each of the two lines is perpendicular to the same line, what kind of lines are they to each other?

Answer:

The figure can be drawn as follows:

Here,and.

We need to find the relation between lines l and m

It is given that , therefore,

(i)

Similarly, we have, therefore,

(ii)

From (i) and (ii), we get:

But these are the pair of corresponding angles.

Theorem states: If a transversal intersects two lines in such a way that a pair of corresponding angles is equal, then the two lines are parallel.

Thus, we can say that .

Hence, the lines are parallel to each other.

Page No 10.47:

Question 9:

In the given figure, ∠1 = 60° and ∠2 = 23rd of a right angle. Prove that l || m.

Answer:

The figure is given as follow:

It is given that

Also,

Thus we have

But these are the pair of corresponding angles.

Thus

Hence proved.

Page No 10.47:

Question 10:

In the given figure, if l || m || n and ∠1 = 60°, find ∠2.

Answer:

The given figure is as follows:

We have and

Thus, we get and as corresponding angles.

Therefore,

(i)

We haveand forming a linear pair.

Therefore, they must be supplementary. That is;

From equation (i):

(ii)

We have

Thus, we get and as alternate interior opposite angles.

Therefore, these must be equal. That is,

From equation (ii), we get :

Hence the required value for is .

Page No 10.47:

Question 11:

Prove that the straight lines perpendicular to the same straight line are parallel to one another.

Answer:

The figure can be drawn as follows:

Here, and.

We need to prove that

It is given that , therefore,

(i)

Similarly, we have, therefore,

(ii)

From (i) and (ii), we get:

But these are the pair of corresponding angles.

Theorem states: If a transversal intersects two lines in such a way that a pair of corresponding angles is equal, then the two lines are parallel.

Thus, we can say that .

Page No 10.47:

Question 12:

The opposite sides of a quadrilateral are parallel. If one angle of the quadrilateral is 60°, find the other angles.

Answer:

Also,and are vertically opposite angles, thus, these two must be equal. That is,

(i)

Also,.

Adding this equation to (i), we get:

But these are the consecutive interior angles which are not supplementary.

Theorem states: If a transversal intersects two lines in such a way that a pair of consecutive interior angles is supplementary, then the two lines are parallel.

Thus, m is not parallel to n.

Page No 10.48:

Question 16:

Which pair of lines in the given figure are parallel? Given reasons.

Answer:

The figure is given as follows:

We haveand.

Clearly,

.

These are the pair of consecutive interior angles.

Theorem states: If a transversal intersects two lines in such a way that a pair of consecutive interior angles is supplementary, then the two lines are parallel.

Thus, .

Similarly, we have and.

Clearly,

.

These are the pair of consecutive interior angles.

Theorem states: If a transversal intersects two lines in such a way that a pair of consecutive interior angles is supplementary, then the two lines are parallel.

Thus,.

Hence the lines which are parallel are as follows:

and .

Page No 10.48:

Question 17:

If l, m, n are three lines such that l || m and n ⊥l. prove that n⊥m.

Answer:

The figure can be drawn as follows:

Here, and

We need to prove that .

It is given that, therefore,

(i)

We have, thus,and are the corresponding angles. Therefore,these must be equal. That is,

From equation (i), we get:

Therefore,.

Hence proved.

Page No 10.48:

Question 18:

Which of the following statements are true (T) and which are false (F)? Give reasons.

(i) If two lines are intersected by a transversal, then corresponding angles are equal.
(ii) If two parallel lines are intersected by a transversal, then alternate interior angles are equal.
(iii) Two lines perpendicular to the same line are perpendicular to each other.
(iv) Two line parallel to the same line are parallel to each other.
(v) If two parallel lines are intersected by a transversal, then the interior angles on the same side of the transversal are equal.

Answer:

(i)

Statement: If two lines are intersected by a transversal, then corresponding angles are equal.

False

Reason:

The above statement holds good if the lines are parallel only.

(ii)

Statement: If two parallel lines are intersected by a transversal, then alternate interior angles are equal.

Statement: Two lines perpendicular to the same line are perpendicular to each other.

False

Reason:

The figure can be drawn as follows:

Here, and

It is given that , therefore,

(i)

Similarly, we have , therefore,

(ii)

From (i) and (ii), we get:

But these are the pair of corresponding angles.

Theorem states: If a transversal intersects two lines in such a way that a pair of corresponding angles is equal, then the two lines are parallel.

Thus, we can say that .

(iv)

Statement: Two lines parallel to the same line are parallel to each other.

True

Reason:

The figure is given as follows:

It is given that and

We need to show that

We have , thus, corresponding angles should be equal.

That is,

Similarly,

Therefore,

But these are the pair of corresponding angles.

Therefore, .

(v)

Statement: If two parallel lines are intersected by a transversal, then interior angles on the same side of the transversal are equal.

False

Reason:

Theorem states: If a transversal intersects two parallel lines then the pair of alternate interior angles is equal.

Page No 10.48:

Question 19:

Fill in the blanks in each of the following to make the statement true:

(i) If two parallel lines are intersected by a transversal, then each pair of corresponding angles are ...
(ii) If two parallel lines are intersected by a transversal, then interior angles on the same side of the transversal are ....
(iii) Two lines perpendicular to the same line are ... to each other.
(iv) Two lines parallel to the same line are ... to each other.
(v) If a transversal intersects a pair of lines in such away that a pair of alternate angles are equal, then the lines are ...
(vi) If a transversal intersects a pair of lines in such away that the sum of interior angles on the same side of transversal is 180°, then the lines are ...

Answer:

(i) If two parallel lines are intersected by a transversal, then corresponding angles are equal.

(ii) If two parallel lines are intersected by a transversal, then interior angles on the same side of the transversal are supplementary.

(iii) Two lines perpendicular to the same line are parallel to each other.

(iv) Two lines parallel to the same line are parallel to each other.

(v) If a transversal intersects a pair of lines in such a way that a pair of interior angles is equal, then the lines are parallel.

(vi) If a transversal intersects a pair of lines in such a way that a pair of interior angles on the same side of transversal is, then the lines are parallel.

Page No 10.48:

Question 20:

In the given figure, AB || CD || EF and GH || KL. Find the ∠HKL.

Answer:

The given figure is as follows:

Let us extend GH to meet AB at Y.

Similarly, extend LK to meet CD at Z.

We have the following:

and are the vertically opposite angles. Therefore,

Since, . Thus,and are the consecutive interior angles.

Therefore,

From (i), we get:

Since,. Thus,and are the corresponding angles.

Therefore,

From (ii), we get:

(iii)

Also,and are the alternate interior opposite angles.

Therefore,

(iv)

Thus, the required angle can be calculated as:

From (iii) and (iv) we get:

Hence, the required value for is.

Page No 10.48:

Question 21:

In the given figure, show that AB || EF.

Answer:

The figure is given as follows:

We need to prove that.

It is given that and

∠ACD=∠ACE+∠ECD∠ACD=22°+35°∠ACD=57°

Thus,

But these are the pair of alternate interior opposite angles.

Theorem states: If a transversal intersects two lines in such a way that a pair of alternate interior angles is equal, then the two lines are parallel.

Therefore,

(i)

It is given that and

Thus,

But these are the pair of consecutive interior opposite angles.

Theorem states: If a transversal intersects two lines in such a way that a pair of consecutive interior angles is supplementary, then the two lines are parallel.